What does a neuron do if it receives inhibitory and excitatory impulses from neighboring neurons?

The answer is that it depends.

There is a process called summation, which determines whether or not an action potential will be generated by the combined effects of excitatory and inhibitory signals. Depending on the total sum of many individual inputs (multiple simultaneous inputs of excitatory and inhibitory), summation may or may not reach the threshold voltage to trigger an action potential.